Functional relations among solutions to the fifth Painlevé equation (Q5951320)

Here, properties of the Bäcklund transformation are studied and nonlinear functional relations among solutions to the fifth Painlevé equation \[ w''= {3w-1\over 2w(w-1)} w^{\prime 2}- {w'\over z}+ {(w-1)^2\over z^2} \Biggl(\alpha w+{\beta\over w}\Biggr)+ {\gamma w\over z}+ {\delta w(w+1)\over w-1} \] are constructed, where \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are arbitrary complex constants. Finally, relationships between solutions to the fifth and third Painlevé equations are analyzed and extended.